Dynamics, synchronization, and quantum phase transitions of two dissipative spins

We analyze the static and dynamic properties of two Ising-coupled quantum spins embedded in a common bosonic bath as an archetype of dissipative quantum mechanics. First, we elucidate the ground-state phase diagram for an Ohmic and a sub-Ohmic bath using a combination of bosonic numerical renormalization group (NRG), analytical techniques, and intuitive arguments. Second, by employing the time-dependent NRG we investigate the system’s rich dynamical behavior arising from the complex interplay between spin-spin and spin-bath interactions. Interestingly, spin oscillations can synchronize due to the proximity of the common non-Markovian bath and the system displays highly entangled steady states for certain nonequilibrium initial preparations. We complement our nonperturbative numerical results by exact analytical solutions when available and provide quantitative limits on the applicability of the perturbative Bloch-Redfield approach at weak coupling.

Figure 1

(Color online) Two quantum spins-$\frac{1}{2}$, ${\mathbf{\sigma }}_1$ and ${\mathbf{\sigma }}_2$, coupled through an Ising interaction $K$. The spins are also entangled, via their ${\sigma }^z$ components, to a common reservoir of bosonic oscillator modes with frequencies ${\omega }_k$. The bath is characterized by the spectral density $J\left(\omega \right)=2\pi \alpha {\omega }^s{\omega }_c^{1-s}\theta \left({\omega }_c-\omega \right)\theta \left(\omega \right)$, where $s=1$ $\left(s<1\right)$ refers to an Ohmic (sub-Ohmic) bosonic environment.

Figure 2

(Color online) Phase diagram of the Ohmic two-spin-boson model as a function of dissipation strength $\alpha$ and Ising coupling $K$. Different curves correspond to different values of tunneling amplitudes ${\Delta }_1={\Delta }_2\equiv \Delta$. For infinitesimal bias fields ${ϵ}_{1,2}=-{10}^{-8}{\omega }_c$ the ground state of the system in the localized region is given by $|\uparrow \uparrow ⟩\otimes |\Omega ⟩$, where $|\Omega ⟩$ is a shifted bath vacuum [see Eq. (6)]. The dashed line indicates where the renormalized Ising interaction vanishes: $K_r=0$.

Figure 3

(Color online) Phase diagram of the sub-Ohmic two-spin-boson model with $s=1/2$ versus $\alpha$ and $K$ and for different values of $\Delta$. The dashed line indicates where $K_r=0$.

Figure 4

(Color online) $⟨{\sigma }_{1,2}^z⟩$ as a function of $\alpha$ for various values of $K$ and $\Delta =0.025{\omega }_c$. Different bias field configurations are shown in the upper part (ferromagnetic, ${ϵ}_1={ϵ}_2={10}^{-8}{\omega }_c$) and lower part (antiferromagnetic, ${ϵ}_1=-{ϵ}_2={10}^{-8}{\omega }_c$) of the figure. This plot shows that spins are always aligned in the localized phase. The expectation values $⟨{\sigma }_{1,2}^z⟩$ remain zero up to larger values of $\alpha$ simply because the antiferromagnetic bias field configuration does not lift the degeneracy of the ground states $\left\{|\uparrow \uparrow ⟩,|\downarrow \downarrow ⟩\right\}$ in the localized phase.

Figure 5

(Color online) Entanglement entropy $\mathcal{E}$ as a function of dissipation $\alpha$ in the Ohmic two-spin-boson model shown for different values of the Ising coupling $K$ and ${\Delta }_{1,2}=0.1{\omega }_c$. The rapid drop to zero around ${\alpha }_c\approx 0.25$ signifies the transition to the localized phase. The plateau for smaller dissipation indicates the loss of phase coherence at $\alpha \approx {\alpha }_c/2$ similar to the single-spin-boson case. The inset shows larger values of $K$ where the (incoherent) plateau shrinks to a peaklike structure, indicating that coherence is lost only right at the phase transition.

Figure 6

(Color online) Entanglement entropy $\mathcal{E}$ as a function of dissipation $\alpha$ for the sub-Ohmic two-spin-boson model with $s=1/2$. Different curves are for different values of the Ising coupling $K$ and ${\Delta }_{1,2}=0.1{\omega }_c$. The entropy $\mathcal{E}$ reaches a maximum at the phase transition (see also Fig. 3) and falls off continuously to both sides of the transition.

Figure 7

(Color online) Scaling of magnetization at the phase transition in the sub-Ohmic system with $s=\frac{1}{2}$. We fit $⟨{\sigma }_1^z⟩$ as a function of the Ising interaction against a power law $⟨{\sigma }_1^z⟩\sim {\left|K-K_c\right|}^{\zeta }$ and find $\zeta =0.5$. Different lines represent fits using $f_i\propto {\left|K-K_c\right|}^{{\zeta }_i}$. Results of the fit and error bars for ${\zeta }_i$, as well as the different values of ${\Delta }_{1,2}$ (in units of ${\omega }_c$) used, are shown in the plot.

Figure 8

(Color online) Comparison of $⟨{\sigma }_{1,2}^x\left(t\right)⟩$ between exact solution in Eq. (23) (dashed) and TD-NRG result (solid) for different values of $\alpha$ and bath exponents $s=1$ (upper part) and $s=\frac{1}{2}$ (lower part). Parameters used are ${ϵ}_{1,2}={\Delta }_{1,2}=0$, $K=0$, ${\omega }_c=1$, and $\alpha$ as specified in the plot.

Figure 9

(Color online) Comparison of the results for $⟨{\sigma }_{1,2}^z\left(t\right)⟩$ from the TD-NRG (solid) and the Bloch-Redfield approach (dashed) for $K={ϵ}_{1,2}=0$, ${\Delta }_{1,2}=0.1{\omega }_c$ in the perturbative regime $\alpha \text{ln}\left({\omega }_c/\Delta \right)⪡1$. Upper (lower) part shows the case $s=1$ $\left(s=\frac{1}{2}\right)$. Deviations between the two solutions are visible already for $\alpha \text{ln}\left({\omega }_c/\Delta \right)=0.01$. Note the beatings in oscillations due to the bath induced Ising coupling.

Figure 10

(Color online) Comparison of results for $⟨{\sigma }_{1,2}^z\left(t\right)⟩$ obtained with the TD-NRG (solid) and the Bloch-Redfield approach (dashed) for $K=0.2{\omega }_c$, ${ϵ}_{1,2}=0$. Other parameters are as in Fig. 9. Deviations between the two solutions are visible already for $\alpha =0.005$.

Figure 11

(Color online) Synchronization of two spins with different spin-flip terms ${\Delta }_2=\frac{1}{2}{\Delta }_1={10}^{-3}{\omega }_c$ by weak coupling to a common bath. There is no direct coupling between the spins $K=0$. The upper part of the figure shows the uncoupled case $\alpha =0$. The middle part shows the Ohmic case and the lower part the sub-Ohmic one with $s=\frac{1}{2}$. We use the same strength $\alpha =8\times {10}^{-4}$ for the different bath dispersions, which lies in the perturbative regime: $\alpha \text{ln}\left({\omega }_c/{\Delta }_2\right)=6\times {10}^{-3}⪡1$.

Figure 12

(Color online) Oscillation amplitudes $A_{\Omega ,\delta }^{\left(1,2\right)}$ as a function of the Ising coupling $K$ [see Eqs. (37, 38)]. Other parameters read ${\Delta }_1=2{\Delta }_2=2\times {10}^{-3}{\omega }_c$. The two-spin expectation values $⟨{\sigma }_{1,2}^z\left(t\right)⟩$ evolve according to Eq. (32).

Figure 13

(Color online) Comparison of spin dynamics between the Ohmic single-spin-boson (thin lines) and Ohmic two-spin-boson models with $K_r=0$ (symbols). Different curves correspond to different values of dissipation strength $\alpha$. The Ising interaction is chosen accordingly to be $K=4\alpha {\omega }_c$. Other parameters are ${\Delta }_{1,2}=\Delta =0.1{\omega }_c$, ${ϵ}_{1,2}=ϵ=0$. The upper panel shows the spin dynamics in the coherent regime $0<\alpha <1/2$. Here, the two-spin oscillations have a slightly larger frequency and are stronger damped than the single-spin oscillations (see also Fig. 15). The lower panel displays the dynamics for stronger dissipation $\alpha \ge 1/2$. For $1/2\le \alpha \le 1$ both systems display incoherent decay. For even larger values of $\alpha$, we observe that, in contrast to the single-spin-boson model which localizes at $\alpha =1$, the two-spin dynamics remains incoherent at least up to $\alpha =1.5$. This is in agreement with the phase diagram of Fig. 2.

Figure 14

(Color online) Comparison of spin dynamics between the sub-Ohmic single-spin (thin lines) and two-spin-boson models with $K_r=0$ (symbols). Different curves correspond to different values of dissipation strength $\alpha$. Ising interaction is chosen accordingly to be $K=8\alpha {\omega }_c$. Other parameters are ${\Delta }_{1,2}=\Delta =0.1{\omega }_c$, ${ϵ}_{1,2}=ϵ=0$. The upper panel shows the spin dynamics for weak dissipation up to $\alpha =0.05$. The two-spin oscillations have slightly larger frequency and are stronger damped compared to the single-spin results. The lower panel shows the case of strong dissipation $\alpha >0.05$. The single-spin-boson model localizes at ${\alpha }_c=0.107$. In contrast, the two-spin-boson model remains delocalized along $K_r=0$, which is in agreement with the phase diagram of Fig. 3.

Figure 15

(Color online) Quality factor of oscillations in the Ohmic (upper panel) and sub-Ohmic (lower panel) single- and two-spin-boson models. TD-NRG results for the single (two-spin) boson model are shown as crosses (open squares). Quality factors derived from the Bloch-Redfield approach (see Sec. 4B) are displayed as filled squares. The solid line denotes the quality factor derived from the NIBA approximation in Eq. (45).

Figure 16

(Color online) Exponential decay of $⟨{\sigma }^z\left(t\right)⟩$ at the Toulouse point $\alpha =1/2$ in the single-spin-boson model. Symbols are TD-NRG results for different tunneling amplitudes $\Delta /{\omega }_c$ which result in different decay rates $\Gamma =\pi {\Delta }^2/2{\omega }_c$. Solid lines are fit with $f_i\left(t\right)=\text{exp}\left[-a_i\Gamma t\right]$ using fit parameters $a_i$ given in Table .

Figure 17

(Color online) Exponential decay of $⟨{\sigma }_{1,2}^z\left(t\right)⟩$ at the generalized Toulouse point $K=2{\omega }_c$, $\alpha =1/2$ in the two-spin-boson model. Symbols are TD-NRG results for different tunneling amplitudes ${\Delta }_{1,2}/{\omega }_c$, which result in different decay rates ${\Gamma }_{1,2}=\pi {\Delta }_{1,2}^2/\left(2{\omega }_c\right)$. Since ${\Delta }_1={\Delta }_2$, we observe ${\Gamma }_1={\Gamma }_2$. Solid lines are fit with $f_i\left(t\right)=\text{exp}\left[-b_i\Gamma t\right]$ using fit parameters $b_i$ given in Table .

Figure 18

(Color online) Spin dynamics $⟨{\sigma }_{1,2}^z\left(t\right)⟩$ for different values of $\alpha$ in the regime of strong spin-bath coupling for an Ohmic (upper part) and a sub-Ohmic bath with $s=1/2$ (lower part). Other parameters read $\Delta =0.1{\omega }_c$, $K=0.2{\omega }_c$, and ${ϵ}_{1,2}=0$. For this choice of $K$ the localization phase transition occurs at ${\alpha }_c\approx 0.25\left(0.043\right)$ in the Ohmic (sub-Ohmic) system.